When a bridge vibrates at its natural frequency, disaster can follow. When a radio tuner resonates, perfect reception follows. Understanding resonance and damping is essential for both engineering design and everyday physics.
π΅Distinguish between free and forced oscillations
πExplain resonance: when driving frequency equals natural frequency
πDescribe light, critical and heavy (overdamped) damping and sketch amplitudeβtime graphs
ποΈEvaluate practical examples where resonance is useful or dangerous
πInterpret resonance curves showing amplitude vs driving frequency for different damping levels
π§Explain how damping affects the resonance frequency and peak amplitude
Free and Forced Oscillations
A free oscillation occurs when a system is displaced and released, oscillating at its own natural frequency fβ without any ongoing external driving force. Examples: a pendulum swinging freely, a plucked guitar string, a child on a swing given one push.
The natural frequency depends on the physical properties of the system:
System
Natural frequency
Mass-spring
fβ = (1/2Ο)β(k/m)
Simple pendulum
fβ = (1/2Ο)β(g/L)
LC circuit (electronics)
fβ = 1/(2Οβ(LC))
A forced oscillation occurs when a periodic driving force is applied at a driving frequency f. The system is compelled to oscillate at f, which may differ from fβ. The amplitude of the forced oscillation depends on how close f is to fβ.
Natural frequency fβ: The frequency at which a system oscillates when freely displaced and released, determined solely by the system's physical properties (mass, stiffness, length etc.).
Resonance
Resonance occurs when the driving frequency equals the natural frequency of the system: f_driving = fβ. At resonance, energy is transferred most efficiently from the driver to the oscillating system, and the amplitude becomes very large (theoretically infinite without damping).
Resonance condition: f_driving = fβ
The amplitude-frequency graph (resonance curve) shows a sharp peak at f = fβ for lightly damped systems. As damping increases:
The peak amplitude decreases
The peak broadens
The resonant peak shifts very slightly below fβ
The sharpness of the resonance peak is described by the Q-factor (quality factor). High Q = sharp resonance (low damping). Low Q = broad resonance (heavy damping). The Q-factor is not on the AQA spec but is useful context.
Examples of Resonance
Useful resonance
Dangerous resonance
Radio/TV tuning circuits β LC resonance selects a station's frequency
Tacoma Narrows Bridge (1940) β wind-driven oscillation at bridge's natural frequency caused collapse
MRI scanners β nuclear magnetic resonance
Mechanical resonance in aircraft engines causing metal fatigue
Musical instruments β resonating air columns
Soldiers breaking step on bridges to prevent resonance
Microwave ovens β water molecules resonating at 2.45 GHz
Earthquake damage to buildings at their natural frequency
Damping
Damping is any mechanism that removes energy from an oscillating system, reducing the amplitude over time. Common damping mechanisms include air resistance, friction, viscous drag (in fluids), and electromagnetic braking.
There are three main categories of damping:
Light damping: The system oscillates with gradually decreasing amplitude. Each successive oscillation has a smaller amplitude than the last. The period is approximately unchanged from the undamped case. Example: a pendulum in air.
Critical damping: The system returns to equilibrium in the shortest possible time without oscillating. This is the optimum damping for many engineering applications. Example: car shock absorbers, door dampers (the door closes quickly without bouncing).
Heavy (over-) damping: The system returns very slowly to equilibrium without oscillating. The return is slower than critically damped. Example: a pendulum submerged in thick oil.
The amplitude of a lightly damped system decreases exponentially with time:
A(t) = Aβ e^(βΞ³t)
where Ξ³ is the damping coefficient (sβ»ΒΉ). The energy decreases as AΒ², so energy β e^(β2Ξ³t) β energy decays twice as fast as amplitude.
Effect of Damping on Resonance
Damping fundamentally changes the resonance behaviour of a system:
No damping: Resonance peak is infinitely tall and infinitely narrow at exactly fβ.
Light damping: Sharp, tall peak very close to fβ. The system is very frequency-selective. Useful in communication circuits where you want to pick out a single frequency.
Moderate damping: Broader, shorter peak. The system responds to a range of driving frequencies. Useful where you need response across a band of frequencies.
Heavy damping: Very broad, flat response. No distinct resonance peak. The system barely oscillates even when driven at fβ.
In engineering design, the choice of damping level depends on the application:
Bridges and buildings: heavy damping to prevent dangerous resonance
Vehicle suspensions: critical or near-critical damping for comfortable ride
Seismographs: light damping so they respond sensitively to earthquake frequencies
Radio tuners: very light damping (high Q) for sharp frequency selectivity
β οΈ Exam trap: heavy damping does NOT eliminate resonance entirely β it just reduces and broadens the peak. The resonant frequency also shifts slightly lower with increasing damping.
Energy Considerations in Damped Oscillations
In a damped system, energy is continuously transferred from the mechanical oscillator to the surrounding medium (as heat, sound, or electrical energy). At resonance in a forced system, the power input from the driver equals the power dissipated by damping β a steady state is reached.
At resonance, the driving force is always in phase with the velocity of the oscillator (not the displacement). This means work done = Fv is always positive β maximum power transfer. At other driving frequencies, the force and velocity are out of phase, reducing the average power input.
In a lightly damped system given a sudden input of energy (an impulse), the resulting "ringing" or decaying oscillation is called a transient. Eventually the system settles to a steady-state forced oscillation at the driving frequency. This transient behaviour is important in electronics, acoustics and control systems.
Real oscillators always have some damping. The "Q-factor" (quality factor) Q β 2Οfβ Γ (energy stored / power dissipated) quantifies how long a system oscillates before its energy decays significantly. High Q β low damping β energy stored for many cycles.
A child on a swing has a natural period of 3.0 s. A parent pushes with a frequency of 0.30 Hz. Will the amplitude grow significantly? Explain your answer.
1Natural frequency: fβ = 1/T = 1/3.0 = 0.333 Hz
2Driving frequency: f = 0.30 Hz
3Since f β fβ (0.30 Hz vs 0.333 Hz), resonance does not occur.
4The forced oscillation amplitude will be relatively small. For resonance, the parent should push at f = 1/3 Hz (every 3.0 s, once per swing).
No β driving frequency (0.30 Hz) does not match natural frequency (0.333 Hz). Push every 3.0 s for resonance.
A lightly damped oscillator has initial amplitude 0.20 m and damping coefficient Ξ³ = 0.15 sβ»ΒΉ. Calculate the amplitude after (a) 5.0 s, (b) 10.0 s. By what factor does the amplitude decrease per second?
2At t = 5.0 s: A = 0.20 Γ e^(β0.75) = 0.20 Γ 0.472 = 0.0944 m β 0.094 m
3At t = 10.0 s: A = 0.20 Γ e^(β1.5) = 0.20 Γ 0.223 = 0.0446 m β 0.045 m
4Per second: factor = e^(βΞ³) = e^(β0.15) = 0.861 β amplitude falls to 86.1% each second.
A(5s) = 0.094 m; A(10s) = 0.045 m; amplitude is 86% of previous value each second.
Explain why car shock absorbers are designed to be critically damped. What would happen if they were (a) underdamped (lightly damped), (b) overdamped (heavily damped)?
1Critical damping: the car returns to its equilibrium ride height as quickly as possible after hitting a bump, without bouncing.
2(a) Underdamped: the car would bounce up and down several times after hitting each bump β uncomfortable ride, reduced steering control, increased wear.
3(b) Overdamped: the car would be very stiff and slow to recover after a bump β a harsh, uncomfortable ride and poor road-holding over successive bumps.
4Critical damping is the ideal compromise: fast return to equilibrium, no oscillation.
Critical damping returns the suspension to equilibrium fastest without oscillation β ideal for passenger comfort and road safety.
A tuning fork of frequency 440 Hz is held near a guitar string. The string vibrates strongly only when its tension is adjusted so that its natural frequency is 440 Hz. Identify the type of oscillation and the phenomenon occurring.
1The tuning fork drives a forced oscillation in the guitar string.
2When the string's natural frequency = 440 Hz = driving frequency, resonance occurs.
3At resonance, energy transfer from fork to string is maximised, giving large amplitude oscillation.
4This is the principle behind sympathetic resonance β used in musical instrument design to amplify specific frequencies.
Forced oscillation leading to resonance when f_string = f_fork = 440 Hz. Maximum energy transfer and maximum amplitude occur at resonance.
Q1. Resonance in a mechanical system occurs when the driving frequency is:
Q2. Which type of damping returns a system to equilibrium in the shortest time without oscillation?
Q3. How does increasing the damping of a resonating system change its resonance curve?
Q4. A soldier marching in step with others across a bridge could be dangerous because:
Q5. In a lightly damped system, successive amplitudes are 10 cm, 8 cm, 6.4 cm ... What type of decrease is this?
Challenge 1. A mass-spring system has natural frequency 5.0 Hz. It is driven by an external force at frequencies of 2.0 Hz, 5.0 Hz and 8.0 Hz in three separate experiments. Sketch (describe) the expected amplitude of oscillation in each case. How does the presence of damping change your answer for the 5.0 Hz case?
β At 2.0 Hz: well below resonance; small, steady forced oscillation amplitude β the system follows the driver sluggishly. At 5.0 Hz: at resonance (f = fβ); amplitude is maximum. Without damping it is theoretically infinite; with damping it is finite but very large, and the system oscillates at 5.0 Hz. The amplitude at resonance is inversely proportional to the damping coefficient. At 8.0 Hz: above resonance; small amplitude forced oscillation β the system cannot keep up with the fast driver. The displacement and driving force are approximately in antiphase above resonance. With damping: the resonance peak at 5.0 Hz is reduced in height and slightly broadened. The resonant frequency shifts very slightly below 5.0 Hz.
Challenge 2. A lightly damped pendulum has initial amplitude 15 cm. After 10 complete oscillations, the amplitude is 9.5 cm. (a) Find the ratio A/Aβ per oscillation. (b) Find the amplitude after 25 oscillations. (c) After how many oscillations will the amplitude first fall below 1 cm?
β (a) Ratio per oscillation r = (A_10/Aβ)^(1/10) = (9.5/15)^(0.1) = (0.6333)^0.1 = e^(0.1 Γ ln0.6333) = e^(0.1 Γ (β0.4575)) = e^(β0.04575) = 0.9553. Each oscillation retains 95.5% of previous amplitude. (b) After 25 oscillations: A = 15 Γ (0.9553)^25 = 15 Γ e^(β25Γ0.04575) = 15 Γ e^(β1.144) = 15 Γ 0.3185 = 4.78 cm. (c) Need 15 Γ (0.9553)^n < 1 β (0.9553)^n < 1/15 = 0.0667 β n Γ ln(0.9553) < ln(0.0667) β n Γ (β0.04575) < β2.707 β n > 59.2. So amplitude first falls below 1 cm after 60 oscillations.
Challenge 3 (Synoptic). An MRI scanner uses nuclear magnetic resonance at radio frequencies around 64 MHz (for 1.5 T field). Explain how the principle of resonance is applied. Why is it important that the RF transmitter frequency matches very precisely the natural precession frequency of hydrogen nuclei? What role does relaxation (damping) play in the MRI signal?
β In MRI, hydrogen nuclei (protons) precess about the external magnetic field at the Larmor frequency f = Ξ³B/(2Ο), where Ξ³ is the gyromagnetic ratio. For 1.5 T, this is ~64 MHz. An RF pulse at exactly this frequency resonantly excites the protons, tipping their magnetisation away from the B-field axis β this is forced oscillation at the natural precession frequency. Precise frequency matching is critical because: (i) even slight mismatch means no energy is absorbed by the protons (the resonance peak is very narrow for lightly damped quantum oscillators); (ii) different tissues have slightly different effective Larmor frequencies; (iii) frequency encodes position in frequency-encoded MRI. After the RF pulse stops, the protons "relax" back to equilibrium, emitting their absorbed energy as an RF signal that is detected. This relaxation is analogous to damping β the T1 and T2 relaxation times characterise how quickly different tissues return to equilibrium, providing the tissue contrast that makes MRI diagnostically valuable.
Challenge 4. A building has a natural frequency of 0.5 Hz. An earthquake produces ground vibrations at 0.4 Hz. A structural engineer proposes fitting tuned mass dampers (a large mass on a spring inside the building). Explain the principle of operation and why the natural frequency of the damper should be tuned to 0.5 Hz.
β The building is currently undergoing forced oscillation at 0.4 Hz β close to but not at its natural frequency of 0.5 Hz. A tuned mass damper (TMD) is a secondary oscillator (large mass on spring/damper system) installed at the top of the building. Its natural frequency is tuned to match the building's natural frequency (0.5 Hz). When the building oscillates, the TMD oscillates in antiphase β it exerts a force on the building that opposes the building's motion, effectively absorbing energy from the building's oscillation and dissipating it as heat through its own dampers. By tuning the TMD to fβ = 0.5 Hz (the building's natural frequency), it is most effective at the frequency where the building is most vulnerable β near resonance. Even though the earthquake is at 0.4 Hz, if the building begins to approach resonance (or during aftershocks at different frequencies), the TMD absorbs energy and prevents dangerously large amplitudes. Famous examples: Taipei 101 (660 tonne TMD), the Millennium Bridge in London (initially had resonance problems; fitted with TMDs).